Problem 39 - Calculus an Applied Approach

Question

Find the standard equation of the sphere. $$ \text { Center: }(-2,1,1) ; \text { tangent to the } x y \text { -plane } $$

Step-by-Step Solution

Answer

The standard equation of the sphere tangent to the xy-plane and centered at (-2,1,1) is \( (x+2)^2+(y-1)^2+(z-1)^2 = 1 \).

Step 1: Finding the Radius

Since the sphere is tangent to the xy-plane at (0,0,0), the z-coordinate of the center of the sphere is the radius. Hence, the radius \(r = |1| = 1\).

Step 2: Formulating the Standard Equation

The standard form of a sphere is \( (x-a)^2+(y-b)^2+(z-c)^2 = r^2 \), where (a, b, c) is the center of the sphere and r is the radius. Here, a = -2, b = 1, c = 1 and r = 1. Substituting these values, the equation becomes: \( (x+2)^2+(y-1)^2+(z-1)^2 = 1 \)

Key Concepts

sphere

A sphere is a perfectly round three-dimensional geometrical object. Imagine a ball or a globe; that's what a sphere looks like. It is the set of all points in space that are equidistant from a given point. This particular point is known as the center of the sphere. The distance from the center to any point on the surface of the sphere is called the radius.

Some key characteristics of a sphere include:

• It has no edges or vertices, unlike a cube.
• All points on the surface are the same distance from the center.
• It is symmetrical from all directions.

In practical terms, knowing how to mathematically describe a sphere is essential for solving geometry problems and understanding spatial relationships in three dimensions.

standard equation of a sphere

The standard equation of a sphere is a simple yet powerful mathematical formula. It helps us represent spheres algebraically in a three-dimensional space. By using this equation, you can easily describe and visualize a sphere.
The standard equation is written as:\[(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2\]
Where:

• (a, b, c) is the center of the sphere.
• r is the radius of the sphere.

When given points like the center and situations like tangency to a plane, you can plug in the specific values for a, b, c, and r into this formula

This allows you to derive the sphere's specific equation, providing a clear picture of its size and position. Understanding this equation is crucial when dealing with spheres in math problems and geometric calculations.

geometry in three dimensions

Geometry in three dimensions expands our understanding of shapes beyond the flat surfaces of two-dimensional geometry. This branch of mathematics involves objects that have depth in addition to height and width. Spheres, cylinders, and cubes are examples of three-dimensional shapes.
Three-dimensional geometry involves several key components:

• Coordinates, which are represented as \(x, y, z\) in a 3D space.
• Planes, which are flat surfaces that extend infinitely in all directions within a three-dimensional area.
• Volume, the amount of space an object occupies, can only be measured due to three-dimensional geometry.

This concept adds a layer of complexity to mathematics by introducing additional axes and dimensions to consider. When working with spheres in three-dimensional geometry, it emphasizes the importance of understanding how objects relate spatially, making calculations such as locating the center and measuring the radius crucial for solving real-world and theoretical problems.